Metropolis(-Hastings) random walk
- proposal in MH is sampled randomly from porpoals/jumping distrib \(J_t(\theta_a|\theta_b)\)
- the accpetance rule/ transition function ratio of densities guarantees convergence
\[
\begin{aligned}
T_t(\theta^t|\theta^{t-1}) = \frac{\frac{p(\theta^*|y)}{J(\theta^*|\theta^{t-1})}}{\frac{p(\theta^{t-1}|y)}{J(\theta^{t-1}|\theta^*)}}
\end{aligned}
\]
- But if steps are poorly chosen converges speed, and accordingly, computational effort can get high
Relation between jumping rule and convergence
\(J(\cdot)\) is only useful jumping distibution, if:
- For any \(\theta\) it is easy to sample from \(J(\theta_a|\theta_b)\)
- It is easy to compute r (e.g. if proposal is symmetric)
- The jump has a reasonable distance
- We don’t reject/ accept jumps to often
(see Gelman et al. (2013))
But how can we know that
Problem
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